While these metrics seem fair for establishing a comparative, it is worth noting that they often lead to mask mistaken estimations. The reason is that the mae, the mse and the mare do not take into account where the estimations have been done in the images. Regarding the detection-based counting approaches, it is possible to exploit the standard metrics commonly used in the detection task that relies on the predicted bounding boxes localizing the objects. We describe more in detail these metrics in sec:back:cnn-based-detectors. However, the problem is more remarkable in the density-based approaches, which provide only a weak localization of the objects.
In order to provide a more accurate evaluation, the authors of [1]} introduced the game metric, suitable for the regression by density estimation counting solutions. Their goal is straightforward: to offer an evaluation metric that simultaneously considers the object count and the estimated locations for the objects. In particular, with the game metric, they suggested to sub-divide the image in \(4^{L}\)  non-overlapping regions and computing the mae in each of these sub-regions. The game can be formulated as:
\(GAME(L) = \frac{1}{N} \sum _{n=1}^{N} (\sum _{l=1}^{4^{L}} |c_{gt}^{l} - c_{pred}^{l}|),\) 
